$ C = \left[\begin{array}{rrr}1 & 5 & 4 \\ 5 & 3 & 5\end{array}\right]$ $ B = \left[\begin{array}{rr}1 & 2 \\ 3 & 1 \\ 3 & 2\end{array}\right]$ What is $ C B$ ?
Answer: Because $ C$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ C B = \left[\begin{array}{rrr}{1} & {5} & {4} \\ {5} & {3} & {5}\end{array}\right] \left[\begin{array}{rr}{1} & \color{#DF0030}{2} \\ {3} & \color{#DF0030}{1} \\ {3} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{1}+{5}\cdot{3}+{4}\cdot{3} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{1}+{5}\cdot{3}+{4}\cdot{3} & ? \\ {5}\cdot{1}+{3}\cdot{3}+{5}\cdot{3} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{1}+{5}\cdot{3}+{4}\cdot{3} & {1}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{2} \\ {5}\cdot{1}+{3}\cdot{3}+{5}\cdot{3} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{1}+{5}\cdot{3}+{4}\cdot{3} & {1}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{2} \\ {5}\cdot{1}+{3}\cdot{3}+{5}\cdot{3} & {5}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{1}+{5}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}28 & 15 \\ 29 & 23\end{array}\right] $